Properties of Real Numbers © Margaux Langenhoven 2014

Properties of Real Numbers: Commutative property Associative property Distributive property Multiplicative Identity property Additive Identity property

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Commutative property Take a moment to calculate the following equations: 4 + 5 + 6 + 7 = 22 5 + 6 + 4 + 7 = 22 7 + 4 + 6 + 5 = 22 6 + 5 + 7 + 4 = 22 The answers are the same! What do you notice?

Commutative property What about these equations? 4 x 5 x 6 x 7 = 840 The answers are the same! What do you notice?

Commutative property Commutative comes from the word “commute”: Which means to... move around!

You may also want to experiment with subtraction and division. Commutative property Use the equations and meaning of commutative to brainstorm what Commutative property entails: You may also want to experiment with subtraction and division.

Commutative property says we can move numbers around and still get the same answer when adding or multiplying.

How could we write this as an algebraic expression? Commutative property of Addition 12 + 5 + 9 = 9 + 12 + 5 How could we write this as an algebraic expression? The rule box will disappear and will leave the algebraic rule. a + b + c = c + a + b HINT: Substitute the numbers with letters. Rule?

How would we write this as an algebraic expression? Commutative property of Multiplication 12 x 5 x 9 = 9 x 12 x 5 How would we write this as an algebraic expression? YOUR TURN Write your own example of an algebraic expression for Commutative property of Multiplication using different letters. a x b x c = c x a x b a · b · c = c · a · b Rule?

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‘Hang out’ or be together! Associative property Associative comes from the word “associate”: Which means to... ‘Hang out’ or be together! For example: Behind each box is an image of a girl. Stacy, Rachel and Tanya go to the mall.

AND If Stacy and Rachel arrive together and meet Tanya there... ...they will all be at the mall. AND If Rachel and Tanya arrive together and meet Stacy there... Behind the boxes, the girls have been grouped differently using brackets. ...they will all be at the mall.

Associative property Therefore... It does not matter how they are grouped, they will all get to the mall.

Associative property works with Addition. Let’s have a look at these equations with numbers: 10 + (5 + 4) = 19 (10 + 5) + 4 = 19 (10 + 4) + 5 = 19 Associative property works with Addition. The answers are the same! What do you notice?

Associative property works with Multiplication as well. And these... 10 x (5 x 4) = 200 (10 x 5) x 4 = 200 (10 x 4) x 5 = 200 Associative property works with Multiplication as well. The answers are the same! What do you notice?

You may also want to experiment with subtraction and division. Associative property Use the equations and meaning of associative to brainstorm what Associative property entails: You may also want to experiment with subtraction and division.

Associative property says we can group numbers in any group and still get the same answer when adding or multiplying.

How could we write this as an algebraic expression? Associative property of Addition (12 + 5) + 9 = (9 + 12) + 5 How could we write this as an algebraic expression? Rule? (a x b) x c = (c x a) x b

How could we write this as an algebraic expression? Associative property of Multiplication (12 x 5) x 9 = (9 x 12) x 5 How could we write this as an algebraic expression? (a x b) x c = (c x a) x b Rule?

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Wait… a(b + c) = ab + ac Distributive property We multiply them! Let’s start Distributive property with the general rule: a(b + c) = ab + ac Before you discuss! What does it mean when letters are joined in algebra? Discuss with your partner what this rule is telling us. Wait… We multiply them!

Distributive property Let’s start Distributive property with the general rule: a(b + c) = ab + ac Feedback? With the Distribution property, we distribute what is attached to the bracket, with what is inside the bracket.

Distributive property Right, now to test it with some numbers: 5(3 + 2) We’ll start by using the Order Of Operations way by calculating the brackets first: 5(3 + 2) = 5(5) = 25

a(b + c) = ab + ac = 25 = 15 + 10 5(3 + 2) = 5(3) + 5(2) Now let’s apply the Distributive property rule to see if we get the same answer: a(b + c) = ab + ac = 25 = 15 + 10 5(3 + 2) = 5(3) + 5(2) It’s the same!

Distributive property With your partner, test the Distributive property rule on another example by replacing the letters with your own numbers: a(b + c) = ab + ac Does it work?

Only + and – inside the bracket! Distributive property says we can distribute what is outside the bracket with what is inside the bracket and multiply them. Only + and – inside the bracket!

How could we write this as an algebraic expression? Distributive property of Addition 3(12 + 5) = 3(9) + 3(2) How could we write this as an algebraic expression? Rule? a(b + c) = a(b) + a(c)

How could we write this as an algebraic expression? Distributive property of Subtraction 3(12 - 5) = 3(9) - 3(2) How could we write this as an algebraic expression? a(b - c) = a(b) - a(c) Rule?

Multiplicative identify property 1 2 63 x 1 = 63 5 x 1 = 5 3 114.3 · 1 = 114.3 The numbered boxes will disappear revealing an example of Multiplicative Identity property. Students need to try to determine what the general rule would be. OR a x 1 = a ∴ ∴ Rule? a · 1 = a

Multiplicative Identity property says we can multiply any number by one and the identity of that number will stay the same.

Additive identify property 1 2 5 + 0 = 5 63 + 0 = 63 3 114.3 + 0 = 114.3 The numbered boxes will disappear revealing an example of Additive Identity property. Students need to try to determine what the general rule would be. ∴ Rule? a + 0 = a

Additive identity property says we can add 0 to any number and the identity of that number will stay the same.